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Choose any three by three square of dates on a calendar page.
Circle any number on the top row, put a line through the other
numbers that are in the same row and column as your circled number.
Repeat this for a number of your choice from the second row. You
should now have just one number left on the bottom row, circle it.
Find the total for the three numbers circled. Compare this total
with the number in the centre of the square. What do you find? Can
you explain why this happens?

Make a set of numbers that use all the digits from 1 to 9, once and
once only. Add them up. The result is divisible by 9. Add each of
the digits in the new number. What is their sum? Now try some other
possibilities for yourself!

It is good to see you explain how you came to your answer. Some
of you said you solved the problem by "trial and error" but I am
sure you made some decisions along the way. Sometimes it is
necessary to try several possible numbers before finding the one
that works for other constraints. Next time try to explain your
thinking. I have included Andrei's solution as one of those that
gave a very complete explanation of how they thought the problem
through.

First I observed that the first digit of the result must be
1:

*

*

4

+

2

8

*

_

_

_

_

1

*

*

*

I used for the other letters the following notation:

a

b

4

+

2

8

c

_

_

_

_

1

d

e

f

2 + a + n = , where n can be 0 or
1.
a + n = 8 + d, where a and d are smaller than 10.

1.1.

n = 0

a = 8 + d.
a can be 8 or 9 and d 0 and 1 respectively, but in the first case 8
was used, and in the second case 0 was used.

1.2.

n = 1

a + 1 = 8 + d
a can be 7, 8 and 9 and d can be 0, 1 and 2. In the first situation
it works because the digits weren't used another time, in the
second combination it doesn't work, and the last situation doesn't
work. The result is:

7

b

4

+

2

8

c

_

_

_

_

1

0

e

f

4 + c =

c can be : 3, 5, 6, 9. Then is: 7,
9, 10, 13. The possibilities are:

c = 5; f = 9

c = 9; f = 3

2.1.

c = 5; f = 9

7

b

4

+

2

8

5

_

_

_

_

1

0

e

9

b + 8 =

Using only 3 and 6 there isn't any possibility.

2.2.

c = 9; f = 3

7

b

4

+

2

8

9

_

_

_

_

1

0

e

3

b + 8 + 1 =

b + 9 =

In this situation only b = 6 and e = 5 satisfies the
condition.

This is the only solution for the problem, because using a
step-by-step method, I obtained only one solution.

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